CHAP. V] EULER'S ^-FUNCTION. 135 6 = 2^+1, c=227+l,... are distinct primes and 2^-f27+...=^ or n—a+1 according as a = 0 or a>0. When q is a prime >3, (/>(x)=2g77 is impossible if p=2qn+l is not prime; while if p is prime it has the two solutions p, 2p. If g = 3 and p is prime, it has the additional solutions 3n+1, 2-3"+1. Next, 4>(x)=2nq is impossible if no one of ^ = 2?i~"g+10 = 0, 1,..., ft —1) is prime and q is not a prime of the form 2*4-1, s = 2x^n; but if q is such a prime or if at least one pv is prime, the equation has solutions of the respective forms bq2, where <£(&) =2n~s; apv, where <£(a) =2*. Finally, <t>(x)==2qr has no solution if p = 2gr4-l is not prime and r^2(j4-l. If p is a prime, but 7*7^2g4-l> the two solutions are pf 2p. If p is not prime, but r = 2^4-1, the two solutions are r2, 2r2. If p is prime and r = 2g+l, all four solutions occur. There is a table of the values n<200 for which cj>(x)—n has solutions. L. Kronecker95 considered two fractions with the denominator m as equivalent if their numerators are congruent modulo m. The number of non-equivalent reduced fractions with the denominator m is therefore <t>(m). If ra = ra'm", where ra', m" are relatively prime, each reduced fraction r/m can be expressed in a single way as a sum of two reduced partial fractions r'/m', r"/m". Conversely, if the latter are reduced fractions, their sum r/m is reduced. Hence <t>(m)=<j>(m'}<$>(mr'}. The latter is also derived (pp. 245-6, added by Hensel) from (4), which is proved (pp. 243-4) by considering the g. c. d. of n with any integer ^n, and also (pp. 266-7) by use of infinite series and products. Proof is given (pp. 300-1) of (5). The Gaussian median value (p. 334) of $(n)/n is 6/?r2 with an error whose order of magnitude is 1/V^j provided we take as the auxiliary number of values of (j)(ri)/n a value of the order of magnitude -\/n logt. n. E. B. Elliott96 considered monomials n = paqb... in the independent variables p, q,.. .. In the expansion of n(l — l/p)m(l — l/#)7h.. ., the aggregate of those monomial terms whose exponents are all ^0 is denoted by Fm(ri). Define ^(prqs...) to be zero if any one of r, s,. .. exceeds 1, but to be ( — I)1 if no one of them exceeds 1, and t of them equal 1. Then (7) Fm^!(n)=SFm((i), Fm+1(n)=3 where d ranges over the monomials pa(f... with 0:ga^< Henceforth, let p, q,... be distinct primes. Then Fi(ri)=(t>(ri), while jFf_1(w) is the sum <r(n) of the divisors of n. In (7), d now ranges over all the divisors of n, and ju(fc) is Merten's function [Inversion]. For m = 0, (72) gives the usual expression for <f>(n)t while (7i) defines <r(ri). For m == 1, (70 becomes (4). If r(1)(n) =r(ri) is the number of divisors d of n, write 95Vorlesungen iiber Zahlentheorie, I, 1901, 125-6. 96Proc. London Math. Soc., 34, 1001, 3-15.